Step 1:

Volume of the balloon $V=\large\frac{4}{3}$$\pi r^3$

Diff w.r.t $t$ we get,

$\large\frac{dv}{dt}=\frac{4}{3}$$\pi 3r^2\large\frac{dr}{dt}$

When $t=0,r=$3 units

When $t=3,r=$6 units

It is given $\large\frac{dv}{dt}$=constant=k(say)

$\therefore k=4\pi r^2\large\frac{dr}{dt}$

Separating the variables we get,

$\therefore kdt=4\pi r^2dr$

Step 2:

Integrate on both sides

$k\int dt=4\pi\int r^2dr$

$kt=4\pi \large\frac{r^3}{3}$$+c$------(1)

When $t=0,r=$3units

$\Rightarrow k(0)=\large\frac{4\pi(3)^3}{3}$$+c$

$c=36\pi$

Step 3:

When $t=3sec,r=6$units

$\Rightarrow k(3)=\large\frac{4\pi(6)^3}{3}$$+36\pi$

$3k=288\pi+36\pi$

$k=\large\frac{324\pi}{3}$

$\;\;\;=108\pi$

$\therefore$ Hence $c=36\pi$ and $k=108\pi$

Step 4:

Now substituting the values in equ(1)

$108\pi t=\large\frac{4}{3}$$\pi r^3+36\pi$

$4\pi r^3=36\pi-108\pi t$

$r^3=9-27t$

$\therefore r=\sqrt[\large 3]{9-27t}$